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Magazine Chapter 1: Problem 53
Use the quadratic formula and a calculator to find all real solutions, correct to three decimals. \(2.232 x^{2}-4.112 x=6.219\)
Short Answer
Expert verified
The real solutions are approximately 2.846 and -0.984.
Step by step solution
01
Convert Equation to Standard Form
First, ensure that the given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). The original equation is \( 2.232x^2 - 4.112x = 6.219 \). To convert this to standard form, subtract 6.219 from both sides to get:\[ 2.232x^2 - 4.112x - 6.219 = 0 \].
02
Identify Coefficients
Identify the coefficients in the standard form equation \( ax^2 + bx + c = 0 \). Here, we have:- \( a = 2.232 \)- \( b = -4.112 \)- \( c = -6.219 \)
03
Apply the Quadratic Formula
The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the values of \( a \), \( b \), and \( c \) into the formula:\[ x = \frac{-(-4.112) \pm \sqrt{(-4.112)^2 - 4 \times 2.232 \times (-6.219)}}{2 \times 2.232} \]
04
Calculate the Discriminant
Calculate the discriminant \( b^2 - 4ac \):\[ (-4.112)^2 - 4 \times 2.232 \times (-6.219) \]This becomes:\[ 16.902144 + 55.463904 = 72.366048 \]
05
Evaluate the Square Root
Evaluate the square root of the discriminant:\[ \sqrt{72.366048} \approx 8.504 \]
06
Find Both Solutions
Now substitute back to find both solutions for \( x \):\[ x = \frac{4.112 \pm 8.504}{4.464} \]Calculate each solution:\[ x_1 = \frac{4.112 + 8.504}{4.464} \approx 2.846 \]\[ x_2 = \frac{4.112 - 8.504}{4.464} \approx -0.984 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
A quadratic equation is a type of polynomial equation of degree two. It is one of the most fundamental concepts in algebra. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where:
- \( a \), \( b \), and \( c \) are constants,
- \( a eq 0 \), ensuring the equation is quadratic rather than linear.
Discriminant
The discriminant is a component of the quadratic formula that provides crucial information about the nature and number of the roots of the quadratic equation. In the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), the expression \( b^2 - 4ac \) is known as the discriminant.
- If the discriminant is positive, the quadratic equation has two distinct real solutions.
- If the discriminant is zero, there is exactly one real solution (also known as a repeated root).
- If the discriminant is negative, there are no real solutions; instead, the equation has two complex solutions.
Real Solutions
Real solutions of a quadratic equation are the x-values where the parabola intersects the x-axis. These solutions are particularly helpful in various practical scenarios such as calculating projectile motion in physics or optimizing business profits.
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), real solutions occur when \( b^2 - 4ac \geq 0 \).
- A positive discriminant results in two solutions, while a zero discriminant results in one solution.
Standard Form
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Expressing an equation in standard form makes it easier to use techniques such as factoring, completing the square, and applying the quadratic formula.
- The first step in solving a quadratic equation is often to rearrange it into standard form.
- Once in standard form, you can easily identify the coefficients \( a \), \( b \), and \( c \), which are essential for plugging into the quadratic formula.
